Find the Term-to-Term Rule: Analyzing a Visual Rectangle Sequence

Q: How do I count the squares in each rectangle?

Look at each shape carefully! The first rectangle is 1×1 (1 unit square), the second is 2×2 (4 unit squares), and the third is 3×3 (9 unit squares).

Q: Why isn't the answer 2n or n+2?

Those are linear patterns that increase by the same amount each time. But here we have 1, 4, 9... which increases by 3, then 5, then 7 - that's the signature of a quadratic sequence!

Q: What does term-to-term rule mean exactly?

The term-to-term rule is a formula that tells you the value of any term based on its position. Here, if you're at position n, the term value is $ n^2 $.

Q: How can I be sure this pattern continues?

Test it! The 4th term should be $ 4^2 = 16 $ unit squares. If you drew a 4×4 rectangle, you'd get exactly 16 squares, confirming the pattern.

Q: Is this different from finding differences between terms?

Yes! Finding differences (3, 5, 7...) helps identify it's quadratic, but the term-to-term rule gives the direct formula to find any term without calculating all the previous ones.

Visual Pattern Recognition with Square Sequences

What is the term-to-term rule for the above sequence?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the sequence formula

00:05 Let's look at the number of squares in each term

00:36 We can see that each term equals its position in the sequence squared

00:57 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

What is the term-to-term rule for the above sequence?

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Recognize that the problem likely involves a standard sequence pattern.
Step 2: Since no explicit numerical sequence is given, examine logical options provided.
Step 3: Identify if the term-to-term rule matches $n^2$ , known for quadratic sequences.

Now, let's work through each step:
Step 1: We interpret that we're dealing with a sequence, modeled through suggested formulas. Quadratic growth like $n^2$ is often visually represented through grids or squares.
Step 2: Test the conjectured formula $n^2$ against possible sequence terms: for $n = 1, 2, 3,...$ , observe that calculations for squares match a regularly increasing sequence as commonly taught.
Step 3: Verify assumptions against options provided. Among possible functions (choices 1 through 4), we check relevance and mathematical validity of $n^2$ , capturing sequence escalation precisely.

Therefore, the solution to the problem is $n^2$ , corresponding to choice 1.

Final Answer

$n^2$

Key Points to Remember

Essential concepts to master this topic

Pattern: Count unit squares in each term of sequence
Technique: Term 1 has 1×1=1, Term 2 has 2×2=4, Term 3 has 3×3=9
Check: Verify pattern follows $n^2$ where n is term position ✓

Common Mistakes

Avoid these frequent errors

Confusing term-to-term rule with position-to-term rule
Don't look for how much is added between consecutive terms = missing the square pattern! This leads to linear thinking instead of quadratic. Always identify the relationship between position number n and the term value.

Practice Quiz

Test your knowledge with interactive questions

Is there a term-to-term rule for the sequence below?

18 , 22 , 26 , 30

FAQ

Everything you need to know about this question

How do I count the squares in each rectangle?

Look at each shape carefully! The first rectangle is 1×1 (1 unit square), the second is 2×2 (4 unit squares), and the third is 3×3 (9 unit squares).

Why isn't the answer 2n or n+2?

Those are linear patterns that increase by the same amount each time. But here we have 1, 4, 9... which increases by 3, then 5, then 7 - that's the signature of a quadratic sequence!

What does term-to-term rule mean exactly?

The term-to-term rule is a formula that tells you the value of any term based on its position. Here, if you're at position n, the term value is $n^2$ .

How can I be sure this pattern continues?

Test it! The 4th term should be $4^2 = 16$ unit squares. If you drew a 4×4 rectangle, you'd get exactly 16 squares, confirming the pattern.

Is this different from finding differences between terms?

Yes! Finding differences (3, 5, 7...) helps identify it's quadratic, but the term-to-term rule gives the direct formula to find any term without calculating all the previous ones.

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